Equations In Divergence Form Research Articles

On correctors for linear elliptic homogenization in the presence of local defects: The case of advection–diffusion

We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6–8]. We have next shown, in our recent work [9], using a slightly different strategy of proof than in our earlier works, that we may also address the equation −aij∂iju=f. The present work is devoted to advection–diffusion equations: −aij∂iju+bj∂ju=f. We prove, under suitable assumptions on the coefficients aij, bj, 1≤i,j≤d (typically that they are the sum of a periodic function and some perturbation in Lp, for suitable p<+∞), that the equation admits a (unique) invariant measure and that this measure may be used to transform the problem into a problem in divergence form, amenable to the techniques we have previously developed for the latter case.

On Lp-Resolvent Estimates for Second-Order Elliptic Equations in Divergence Form

We consider the Dirichlet problems for second-order linear elliptic equations in divergence form. The leading coefficient A has small BMO semi-norm and first-order coefficient b belongs to Lr, where $n \leq r < \infty $ if n ≥ 3 and $2 < r < \infty $ if n = 2. We first establish Lp-resolvent estimates on bounded domains having small Lipschitz constant when $r/(r-1) < p < \infty $ . Under the additional assumption div A ∈ Lr, we also establish Lp-resolvent estimates on bounded domains with C1,1 boundary when 1 < p < r.

Green's function for elliptic systems: Moment bounds

We study estimates of the Green's function in $\mathbb{R}^d$ with $d ≥ 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d ≥ 3$, we obtain estimates on the Green's function, its gradient, and the second mixed derivatives which scale optimally in space, in terms of the 'minimal radius' $r_*$ introduced in [Gloria, Neukamm, and Otto: A regularity theory for random elliptic operators; ArXiv e-prints (2014)]. As an application, our result implies optimal stochastic Gaussian bounds on the Green's function and its derivatives in the realm of homogenization of equations with random coefficient fields with finite range of dependence. In two dimensions, where in general the Green's function does not exist, we construct its gradient and show the corresponding estimates on the gradient and mixed second derivatives. Since we do not use any scalar methods in the argument, the result holds in the case of uniformly elliptic systems as well.

Local Hölder estimates for non-uniformly variable exponent elliptic equations in divergence form

In this paper we obtain the local Hölder regularity of the gradients of weak solutions for a class of non-uniformly nonlinear variable exponent elliptic equations in divergence formincluding the following special modelunder some proper assumptions on Ai and the Hölder continuous functions f, pi(x) for i = 1, 2.

On $$C^1$$ C 1 , $$C^2$$ C 2 , and weak type-(1, 1) estimates for linear elliptic operators: part II

We extend and improve the results in \cite{DK16}: showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form equations. We extend the weak type-(1, 1) estimates in \cite{DK16} and \cite{Es94} up to the boundary and derive a Harnack inequality for non-negative adjoint solutions to non-divergence form elliptic equations, when the leading coefficients have Dini mean oscillation.

Optimal elliptic regularity: A comparison between local and nonlocal equations

Given $L≥1$ , we discuss the problem of determining the highest $α=α(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^α_{\rm loc}$ . This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $α(L)≳ {\rm exp}(-CL^β)$ , for some $C, β≥q$ depending on the dimension $N≥q$ . We show that in the non-local case, $α(L)≳ L^{-1-δ}$ for all $δ&gt;0$ .

Besov regularity for solutions of p-harmonic equations

Abstract We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., {\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when {\mathcal{A}} is a p-harmonic type operator, and under the assumption that {x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space {B^{\alpha}_{{n/\alpha},q}} . We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When {\operatorname{div}F=0} , we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map {x\mapsto\mathcal{A}(x,\xi\/)} .

On the initial-boundary value problem for some quasilinear parabolic equations of divergence form

In this paper we give an existence theorem of global classical solution to the initial boundary value problem for the quasilinear parabolic equations of divergence form ut−div{σ(|∇u|2)∇u}=f(∇u,u,x,t) where σ(|∇u|2) may not be bounded as |∇u|→∞. As an application the logarithmic type nonlinearity σ(|∇u|2)=log⁡(1+|∇u|2) which is growing as |∇u|→∞ and degenerate at |∇u|=0 is considered under f≡0.

Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences

We give a derivation of the Vlasov–Maxwell and Vlasov– Poisson–Poisson equations from the Lagrangians of classical electrodynamics. The equations of electromagnetic hydrodynamics (EMHD) and electrostatics with gravitation are derived from them by means of a `hydrodynamical' substitution. We obtain and compare the Lagrange identities for various types of Vlasov equations and EMHD equations. We discuss the advantages of writing the EMHD equations in Godunov's double divergence form. We analyze stationary solutions of the Vlasov– Poisson– Poisson equation, which give rise to non-linear elliptic equations with various properties and various kinds of behaviour of the trajectories of particles as the mass passes through a critical value. We show that the classical equations can be derived from the Liouville equation by the Hamilton–Jacobi method and give an analogue of this procedure for the Vlasov equation as well as in the non-Hamiltonian case.

Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a $C^{1,1}$ regularity theory down to microscopic scale, which is of independent interest and is inspired by the $C^{0,1}$ theory introduced in the divergence form case by the first author and Smart \cite{AS2}.

Global [formula omitted]-estimates and Hölder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions

We study the elliptic equations in divergence form with the general nonstandard growth conditions involving Lebesgue measurable functions on Ω. In this paper we prove global L∞-estimates and Hölder continuity of weak solutions for these equations. Our results improve the one in Giusti (2003) even for the case when p(x)≡p and q(x)≡q are constants.

Nash Estimates and Upper Bounds for Non-homogeneous Kolmogorov Equations

We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and generalizes some classical results by Nash, Aronson and Davies. The class considered has relevant applications in the theory of stochastic processes, in physics and in mathematical finance.

On C1, C2, and weak type-(1,1) estimates for linear elliptic operators

ABSTRACTWe show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.

Integrability and continuity of solutions to double divergence form equations

We obtain sharp conditions for higher integrability and continuity of solutions to double divergence form second-order elliptic equations with coefficients of low regularity. In addition, we prove Harnack’s inequality in this case.

High order correctors and two-scale expansions in stochastic homogenization

In this paper, we study high order correctors in stochastic homogenization. We consider elliptic equations in divergence form on $$\mathbb {Z}^d$$ , with the random coefficients constructed from i.i.d. random variables. We prove moment bounds on the high order correctors and their gradients under dimensional constraints. It implies the existence of stationary correctors and stationary gradients in high dimensions. As an application, we prove a two-scale expansion of the solutions to the random PDE, which identifies the first and higher order random fluctuations in a strong sense.

The additive structure of elliptic homogenization

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in \cite{AKM}: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

Regularity of Weak Solutions to a Model Problem with Conjugation Conditions for Quasilinear Parabolic Systems of Equations

We consider a parabolic quasilinear second order system of equations in divergence form in a model parabolic cylinder. We prove the Holder continuity of a weak solution on a set of full measure in the cylinder. It is shown that the linear system has no singular set. We use a modified method of A-caloric approximation which takes into account the conjugation conditions on the interface between media.

Existence and uniqueness of solutions of hyperbolic equations in divergence form with various boundary conditions on various parts of the boundary

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.

Nonlinear elliptic systems and mean-field games

We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton–Jacobi–Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of N-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some N-person games.

A note on the Harnack inequality for elliptic equations in divergence form

In 1957, De Giorgi [3] proved the H\"{o}lder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in [3].